Featured Poker Blog Post by jajvirta

Thoughts about winrates and variance

Yeah, you heard it first from me: decrease your variance by increasing your winrate.

A couple of disclaimers. First, "variance" is actually not a "thing." I guess you should call the "thing" dispersion. Variance is just a measure of that. "Decreasing variance" would be something like "making kilograms bigger" if you're trying to lose weight or something. Second, what is commonly thought of as variance is the possibility of losing in a given set of hands even if you're a winning player. So it's not like anyone's complaining of the variance of a game where you win either $10000 or $20000 even though the variance is huge in such a game. Third, we can't actually decrease the variance, so to speak, without compromising our EV. What most people want is to increase the probability that one's up after a given set of hands.

But anyhow.

I bet you were disappointed when you read the first sentence. It's not like you can increase your winrate on will. The point, however, is that there are ways to decrease your winrate and not doing those will "decrease your variance", so to speak. Again, not anything terribly surprising here.

What prompted me to write about this, then, was that some players advocate the notion that you should increase the amount of hands played to deal with swings. That is, if a winning player can have a 100k breakeven stretch, then the proposed cure is to play so much that 100k hand period so short that you don't have "long lasting" downswings. And it's true too: if you manage to play million hands with the winrate of 1BB/100, it's pretty much absolutely certain that you are up.

But increasing the amount of hands comes with a cost. Presumably you're already playing as much as possible, given the circumstances. You might even think that you should spend more time improving your game instead of playing all the time. (Or, say, hang out with your family. :-) Increasing the amount of tables might not be an option, as you might already have trouble with the four tables you've figured to be feasible.

And some people, *cough coaches cough*, have the chutzpah to claim that you should probably play less tables. "I might have a downswing that lasts a whole year that way," you might yell.

It's not that simple though.

Let's define some notions first. We have a sample of N hands with the winrate of WR which yields the mean expectation, ME. The result of a hand has a standard deviation of X, which we set here to be 1.8BB. Typical values of standard deviation for 100 hands in short-handed limit hold'em is somewhere around 16-18BB/100. (You can find out yours from Poker Tracker's Session tab. There's "More detail" button there or something.) Standard deviation of N hands is standard deviation of a hand times square root of the amount of hands.

Now, we assume that the results are normally distributed. They pretty much are, but it's worth noting that that's what we assume. Next, we plant the normal distribution curve on the spot where our expected result is after N hands. If we play 1000 hands with the winrate of 1BB/100 then the most likely result for us is 10BB. With this method, we can find out what's the probability that we're not losing after a set of hands. Now that we've planted the distribution on the center of the expected value, we just calculate the area of the curve on the left side of the zero result. And since we know the standard deviation of the sample (std dev of a hand * square root of the number of hands) we know the size and the shape of the curve.

Sounds a bit complicated and it is too, in fact. It involves funny looking characters and hairy derivations. But luckily it's all thought out already and we can just enjoy the results. So, let's take a quick example. We have a player with winrate of 1BB/100 and std dev of 1.8 per hand (typical for lower stakes short handed limit hold'em). We ask whether the result is positive after 1000 hands.

So, first, we have the mean value, the expected value,

0.01BB per hand * 1000 = 10BB

Next, we calculate the standard deviation of the sample:

1.8 per hand * sqrt(1000) = 56.92

Then we find the z-zero point, that is, where the zero value is in the curve.

z0 = (0 - mean) / std dev of the sample

or

(0 - 10) / 56.92 = -0.18

In other words, one fifth of a standard devation from the mean. Now we can calculate the area what falls left of this point. To do this, consult a "z-score probability table", for example here: http://www.statsoft.com/textbook/stathome.html?sttable.html&1

this equals to something like 43% (or, 50 - 7). You will see from the table that 0.18 equals to 0.07 (or 7%). This the area from the mean to this point. So the area from this point to the extreme left (infinity) is approximately 43%.

The other way is to use spreadsheet program and use the function NORMSDIST.

So the probability of us being up after 1000 hands with winrate of 1BB/100 and std dev of 1.8 per hand is staggering 57%.

Now, if we expand this idea to playing 5000 hands per month and considering playing a lot more to "decrease the variance", or at least to decrease the probability of being down. Let's say you add a table to your multi-tabling setup and maybe play bit longer sessions. Doubling the amount of hands without adding more time would of course mean doubling the amount of tables you play, which might be infeasible if you already play 4 tables at a time.

Let's further say that your winrate with less tables is 1.5BB/100 and with more tables it drops to 1.0BB/100. The expected value of 5k hand stretch with less tables compared to 10k hands with more is of course a bit lower, 75BB and 100BB respectively. But the funny thing is that the probability that you're on the positive side (that is, winning) is almost the same with both scenarios, 5k and 1.5 winrate and 10k and 1.0 winrate, 71-72%. You can put this calculation in your spreadsheet and compare different scenarios and see for yourself how much the winrate actually affects the variance.

We can of course debate whether playing less tables increases your winrate at all, and if, how much, but I'm personally pretty sure it does, and often pretty significantly so. This little post was intended to show that increasing your winrate does have a significant effect on the aspect of poker which most players find repulsive: losing money while playing better than opponents.

As a final note, what prompted to think about the whole issue was the fact that people tend to worry most about the "poker tracker result". That is, there are sites that take a lot of rake but give some of it back with rakebacks and bonuses. The true winrate is a function of the rakeback, but people want the poker tracker to show green also. But with rakes as high as 4-5BB/100, you are going to have tough going with this goal. If your true non-rake-deducted winrate with this scenario is something like 5BB/100, which is a good winrate I might add, then your "poker tracker winrate" is 1BB/100, which is going to have huge swings and you might expect to have break-even stretches of tens of thousands of hands.

jajvirta

my english or may brain... maybe both.. . so hard to understand for me

Heh, it's not the clearest explanation, now that I've re-read it myself. But it might be partly that it implicitly assumes some knowledge on the basic statistics and if you don't have much, it might not make a lot of sense because of that.

DavidC

Firstly, from his forum posts, Jaj pretty much rocks, so I don't have anything against him, personally. However, I don't like this post, really. I'm a nit.

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I'm not sure, but I think when you say "decrease your variance" you mean "decrease your risk" (i.e. reducing var/ev, rather than reducing var).

If you were actually talking about decreasing your variance itself by increasing winrate, I think you'd be discussing stuff like how winners might get put in less marginal spots than losers due to their opponents being more intimidated, or something like that. I feel that situation would reduce actual variance, not just risk, but I'm not totally sure.

Also, you seem to be defining variance as deviation from break-even over a given sample, where you should be defining it as deviation from expectation. If you have a 2/100 winrate and play 10,000 hands, a break-even stretch actually represents a 200bb downswing from your expectation. It's kinda interesting that when you think of people's commonly described massive downswings as being stuff like -500bb (from break-even) over 20,000 hands, that swing size doesn't even include their expectation over that sample, which means it's even bigger than we initially thought!

Finally, I'm curious about how to set my bankroll size to give myself 0.1% ROR over an infinite sample... if anyone knows. Although I suppose determining ROR with a given bankroll size would work too, I could just play with the numbers until I got the right ROR.

DJ Sensei

Good post jaj. The toughest part, of course, is figuring out exactly what our winrates are, given our generally small sample sizes, high standard deviations, and the fact that our games are constantly changing. Perhaps your article could be accompanied with a discussion of confidence intervals for winrates over N hands.

jajvirta

I'm not sure, but I think when you say "decrease your variance" you mean "decrease your risk" (i.e. reducing var/ev, rather than reducing var).

Yeah, I tried to explain that in the second paragraph, but I guess I hurried over too quickly. I agree with you completely and I think this a common confusion. But after that second pararagraph, I just translated "decrease the risk of being down after a set of hands" to "decrease your variance", of which the latter is somewhat non-sensical as you point out too. But I do think that what most people want is to not lose money and this what was I aiming to talk about in the article.

DavidC

I'm not sure, but I think when you say "decrease your variance" you mean "decrease your risk" (i.e. reducing var/ev, rather than reducing var).

Yeah, I tried to explain that in the second paragraph, but I guess I hurried over too quickly. I agree with you completely and I think this a common confusion. But after that second pararagraph, I just translated "decrease the risk of being down after a set of hands" to "decrease your variance", of which the latter is somewhat non-sensical as you point out too. But I do think that what most people want is to not lose money and this what was I aiming to talk about in the article.

PygmyHero

Hey jajvirta, you know I was a big fan of this when you originally posted it in the Beginner LHE forum, but I just wanted to give you some dap here too - really good work. Thanks for taking the time to put it together.